Example 1 Points and Lines Example 2 Use Postulates Example 3 Write a Paragraph Proof Lesson 5 Contents

Plan Draw a diagram of a hexagon to illustrate the solution. SNOW CRYSTALS Some snow crystals are shaped like regular hexagons. How many lines must be drawn to interconnect all vertices of a hexagonal snow crystal? Explore The snow crystal has six vertices since a regular hexagon has six vertices. Plan Draw a diagram of a hexagon to illustrate the solution. Example 5-1a

Solve. Label the vertices of the hexagon A, B, C, D, E, and F Solve Label the vertices of the hexagon A, B, C, D, E, and F. Connect each point with every other point. Then, count the number of segments. Between every two points there is exactly one segment. Be sure to include the sides of the hexagon. For the six points, fifteen segments can be drawn. Example 5-1b

Examine In the figure,. are all segments Examine In the figure, are all segments that connect the vertices of the snow crystal. Answer: 15 Example 5-1b

ART Jodi is making a string art design ART Jodi is making a string art design. She has positioned ten nails, similar to the vertices of a decagon, onto a board. How many strings will she need to interconnect all vertices of the design? Answer: 45 Example 5-1c

If plane T contains contains point G, then plane T contains point G. Determine whether the following statement is always, sometimes, or never true. Explain. If plane T contains contains point G, then plane T contains point G. Answer: Always; Postulate 2.5 states that if two points lie in a plane, then the entire line containing those points lies in the plane. Example 5-2a

For , if X lies in plane Q and Y lies in plane R, then plane Q intersects plane R. Determine whether the following statement is always, sometimes, or never true. Explain. Answer: Sometimes; planes Q and R can be parallel, and can intersect both planes. Example 5-2b

contains three noncollinear points. Determine whether the following statement is always, sometimes, or never true. Explain. contains three noncollinear points. Answer: Never; noncollinear points do not lie on the same line by definition. Example 5-2c

a. Plane A and plane B intersect in one point. Determine whether each statement is always, sometimes, or never true. Explain. a. Plane A and plane B intersect in one point. b. Point N lies in plane X and point R lies in plane Z. You can draw only one line that contains both points N and R. Answer: Never; Postulate 2.7 states that if two planes intersect, then their intersection is a line. Answer: Always; Postulate 2.1 states that through any two points, there is exactly one line. Example 5-2d

c. Two planes will always intersect a line. Determine whether each statement is always, sometimes, or never true. Explain. c. Two planes will always intersect a line. Answer: Sometimes; Postulate 2.7 states that if the two planes intersect, then their intersection is a line. It does not say what to expect if the planes do not intersect. Example 5-2e

Given intersecting , write a paragraph proof to show that A, C, and D determine a plane. Given: intersects Prove: ACD is a plane. Proof: must intersect at C because if two lines intersect, then their intersection is exactly one point. Point A is on and point D is on Therefore, points A and D are not collinear. Therefore, ACD is a plane as it contains three points not on the same line. Example 5-3a

Given is the midpoint of and X is the midpoint of write a paragraph proof to show that Example 5-3b

Proof: We are given that S is the midpoint of and X is the midpoint of By the definition of midpoint, Using the definition of congruent segments, Also using the given statement and the definition of congruent segments, If then Since S and X are midpoints, By substitution, and by definition of congruence, Example 5-3c

End of Lesson 5

Example 1 Verify Algebraic Relationships Example 2 Write a Two-Column Proof Example 3 Justify Geometric Relationships Example 4 Geometric Proof Lesson 6 Contents

Algebraic Steps Properties Solve Algebraic Steps Properties Original equation Distributive Property Substitution Property Addition Property Example 6-1a

Substitution Property Division Property Substitution Property Answer: Example 6-1b

Algebraic Steps Properties Solve Algebraic Steps Properties Original equation Distributive Property Substitution Property Subtraction Property Example 6-1c

Substitution Property Division Property Substitution Property Answer: Example 6-1d

Write a two-column proof. then Proof: If Write a two-column proof. then Statements Reasons Proof: 1. Given 1. 2. 2. Multiplication Property 3. 3. Substitution 4. 4. Subtraction Property 5. 5. Substitution 6. 6. Division Property 7. 7. Substitution Example 6-2a

Write a two-column proof. If then Statements Reasons 1. Given 1. 2. Multiplication Property 2. 3. Distributive Property 3. 4. Subtraction Property 4. 5. Substitution 5. 6. Addition Property 6. Example 6-2c

Write a two-column proof. If then Statements Reasons 7. Substitution 7. 8. Division Property 8. 9. Substitution 9. Example 6-2e

Write a two-column proof for the following. Example 6-2f

2. Multiplication Property 1. Given 2. Multiplication Property 3. Substitution 4. Subtraction Property 5. Substitution 6. Division Property 7. Substitution Proof: Statements Reasons 1. 2. 3. 4. 5. 6. 7. Example 6-2g

Write a two-column proof for the following. Prove: b. Given: Write a two-column proof for the following. Example 6-2h

2. Multiplication Property Proof: Statements Reasons 1. Given 2. Multiplication Property 3. Distributive Property 4. Subtraction Property 5. Substitution 6. Subtraction Property 7. Substitution 1. 2. 3. 4. 5. 6. 7. Example 6-2i

A I only B I and II C I and III D I, II, and III MULTIPLE- CHOICE TEST ITEM then which of the following is a valid conclusion? I II III If and Read the Test Item Determine whether the statements are true based on the given information. Example 6-3a

Solve the Test Item Statement I: Examine the given information, GH JK ST and . From the definition of congruence of segments, if , then ST RP. You can substitute RP for ST in GH JK ST to get GH JK RP. Thus, Statement I is true. Statement II: Since the order you name the endpoints of a segment is not important, and TS = PR. Thus, Statement II is true. Example 6-3b

Statement III If GH JK ST, then . Statement III is not true. Because Statements I and II only are true, choice B is correct. Answer: B Example 6-3c

MULTIPLE- CHOICE TEST ITEM If and then which of the following is a valid conclusion? I. II. III. MULTIPLE- CHOICE TEST ITEM A I only B I and II C I and III D II and III Answer: C Example 6-3d

SEA LIFE A starfish has five legs SEA LIFE A starfish has five legs. If the length of leg 1 is 22 centimeters, and leg 1 is congruent to leg 2, and leg 2 is congruent to leg 3, prove that leg 3 has length 22 centimeters. Given: m leg 1 22 cm Prove: m leg 3 22 cm Example 6-4a

3. Definition of congruence m leg 1 m leg 3 3. Proof: Statements Reasons 1. Given 1. 2. Transitive Property 2. 3. Definition of congruence m leg 1 m leg 3 3. 4. Given m leg 1 22 cm 4. 5. Transitive Property m leg 3 22 cm 5. Example 6-4a

DRIVING A stop sign as shown below is a regular octagon DRIVING A stop sign as shown below is a regular octagon. If the measure of angle A is 135 and angle A is congruent to angle G, prove that the measure of angle G is 135. Example 6-4b

3. Definition of congruent angles Proof: Statements Reasons 1. Given 2. Given 3. Definition of congruent angles 4. Transitive Property 1. 2. 3. 4. Example 6-4c

End of Lesson 6

Example 1 Proof With Segment Addition Example 2 Proof With Segment Congruence Lesson 7 Contents

3. Segment Addition Postulate PR – QR = PQ; QS – QR = RS 3. Given: PR = QS Prove the following. Prove: PQ = RS Proof: Statements Reasons 1. Given PR = QS 1. 2. Subtraction Property PR – QR = QS – QR 2. 3. Segment Addition Postulate PR – QR = PQ; QS – QR = RS 3. 4. Substitution PQ = RS 4. Example 7-1a

Prove the following. Prove: Given: Example 7-1b

5. Segment Addition Property AC + CY = AY; BX + XD = BD Proof: Statements Reasons 1. Given 2. Transitive Property 3. Given 4. Addition Property AC = AB, AB = BX AC = BX CY = XD AC + CY = BX + XD 5. Segment Addition Property AC + CY = AY; BX + XD = BD AY = BD 6. Substitution 1. 2. 3. 4. 5. 6. Example 7-1c

Prove the following. Prove: Given: Example 7-2a

2. Definition of congruent segments 2. Proof: Statements Reasons 1. Given 1. 2. Definition of congruent segments 2. 3. Given 3. 4. Transitive Property 4. 5. Transitive Property 5. Example 7-2b

Prove the following. Prove: Given: Example 7-2c

Proof: Statements Reasons 1. Given 2. Transitive Property 3. Given 5. Symmetric Property 1. 2. 3. 4. 5. Example 7-2d

End of Lesson 7

Example 1 Angle Addition Example 2 Supplementary Angles Example 3 Use Supplementary Angles Example 4 Vertical Angles Lesson 8 Contents

TIME At 4 o’clock, the angle between the hour and minute hands of a clock is 120º. If the second hand stops where it bisects the angle between the hour and minute hands, what are the measures of the angles between the minute and second hands and between the second and hour hands? If the second hand stops where the angle is bisected, then the angle between the minute and second hands is one-half the measure of the angle formed by the hour and minute hands, or . Example 8-1a

By the Angle Addition Postulate, the sum of the two angles is 120, so the angle between the second and hour hands is also 60º. Answer: They are both 60º by the definition of angle bisector and the Angle Addition Postulate. Example 8-1b

and is a right angle, find QUILTING The diagram below shows one square for a particular quilt pattern. If and is a right angle, find Answer: 50 Example 8-1b

form a linear pair and find If and Supplement Theorem Subtraction Property Answer: 14 Example 8-2a

are complementary angles and . and If find Answer: 28 Example 8-2b

In the figure, form a linear pair, and Prove that are congruent. and Given: form a linear pair. Prove: Example 8-3a

2. Linear pairs are supplementary. 2. Proof: Statements Reasons 1. Given 1. 2. Linear pairs are supplementary. 2. 3. Definition of supplementary angles 3. 4. Subtraction Property 4. 5. Substitution 5. 6. Definition of congruent angles 6. Example 8-3b

In the figure, NYR and RYA form a linear pair, AXY and AXZ form a linear pair, and RYA and AXZ are congruent. Prove that RYN and AXY are congruent. Example 8-3c

2. If two s form a linear pair, then they are suppl. s. Proof: Statements Reasons 1. Given 2. If two s form a linear pair, then they are suppl. s. 3. Given 4. 1. 2. 3. linear pairs. Example 8-3d

If 1 and 2 are vertical angles and m1 and m2 find m1 and m2. Vertical Angles Theorem 1 2 Definition of congruent angles m1 m2 Substitution Add 2d to each side. Add 32 to each side. Divide each side by 3. Example 8-4a

Answer: m1 = 37 and m2 = 37 Example 8-4b

If and are vertical angles and and find and If and are vertical angles and and Answer: mA = 52; mZ = 52 Example 8-4b

End of Lesson 8

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